Poincare coordinates on anti-de Sitter spacetime ($d{s}^2=z^{-2}(d{z}^2+{\eta }_{ab}d{u}^{a}d{u}^b)$)

For Lorentzian boundary metric $\eta$ and $z>0$, this Anti-de Sitter spacetime patch has unit curvature radius. Its orthonormal coframe has connection ${\omega }^a{}_3=-{\theta }^a$ and curvature ${\Omega }^a{}_b=-{\theta }^a\wedge {\theta }_b$, giving sectional curvature $-1$. The full local isometry algebra is $\mathfrak{so}(d-1,2)$ in spacetime dimension $d$. A patch need not support every local Killing flow for all parameter values.

For unit-radius metric $d{y}^2+e^{-2y}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$, translations conserve $p^{\mu }=e^{-2y}{\dot{x}}^{\mu }$. Timelike normalization gives $K^2=-{\eta }_{\mu \nu }{p}^{\mu }{p}^{\nu }>0$ and ${\dot{y}}^2=K^2{e}^{2y}-1$. Setting $w=e^{-y}$ yields ${\dot{w}}^2+w^2=K^2$, hence the displayed proper-time solution where the cosine is positive. Its two patch-horizon crossings are separated by proper time $\pi$, although the globally extended geodesic is complete.

The boundary of a chosen Poincare coordinates on anti-de Sitter spacetime patch inside the full spacetime is a null coordinate horizon. It is not the timelike conformal boundary, and not a curvature singularity. Timelike geodesics can cross it in finite proper time while the patch time diverges. In a signed-diameter conformal strip, the patch containing $T=\rho =0$ satisfies $|T|<\pi /2+\rho$.